CIE A Level Maths: Pure 3复习笔记1.3.2 Squared Linear Denominators

Partial fractions

• In partial fractions the common denominator is split into parts (factors)
• This is the reverse process to adding (or subtracting) fractions
• In harder questions there is a repeated factor, this is a squared linear factor

What are squared linear denominators?

• A linear factor is of the form (ax + b)
• It is possible b = 0 so a linear factor could be of the form ax (eg 4x)
• A squared linear factor is of the form (ax + b)²
• With b = 0 this would be of the form (ax)² (x² would be too!)

How do I find partial fractions with squared linear denominators?

STEP 1        Factorise the denominator (Sometimes the numerator can be factorised too)

STEP 2        Split the fraction into a sum with a squared linear denominator and any other single linear denominators

STEP 3        Multiply by the denominator to get rid of fractions

STEP 4        Substitute values of x to find A, B, etc (or use comparing coefficients)

STEP 5        Write the original as partial fractions